Beating an old horse .(9) question

[CjStaal]

I know, and conform to the ideal that .9 infinitely repeating does in fact equal 1 but I have a thought that keeps bothering me.
How can.9999999999999999999999999999999999999999999999999999999(repeating forever) multiplied by 2, equal 2?

 

[Mensanator]

I know, and conform to the ideal that .9 infinitely repeating does in fact equal 1 but I have a thought that keeps bothering me.
How can.9999999999999999999999999999999999999999999999999999999(repeating forever) multiplied by 2, equal 2?

What do you think it should be?

 

[dodo]

Is that the right question? Because, if .9999... equals 1, then .9999... multiplied by 2 equals 1 multiplied by 2.

Maybe you're trying to ask how is multiplication of periodic decimals defined; or if it is defined at all, other than by the procedure in the first paragraph. (Not that I know the answer, but I'm interested in the question.)

Edit: sorry, I was answering the original post, not the second; the second poster was (again) faster than me.

 

[CjStaal]

Well I guess look at it this way. .(9)^2 =1? while 9^2 =81... That just confuses me.

 

[dodo]

It's an optical illusion. :) You may want to try a few more nines, like 99^2, 999^2 or 99999999^2.

 

[CjStaal]

Yea, they always execute with the length of the original number but with 8 at the end then the length of the number, with a 1 at the end terminating it.
ie. 9*9=81
99*99=9801
999*999=998001
9999*9999=99980001
99999*99999=9999800001

 

[dodo]

And this decoration looks like an artifact caused by the number of nines being finite.

 

[Hurkyl]

0.99 * 0.99 is less than 0.99(9) * 0.99(9), right?

But 0.99(9) * 0.99(9) cannot be greater than 0.99(9), right?

So, can you tell me what the first digit after the decimal point of 0.99(9) * 0.99(9) is?



0.999 * 0.999 is less than 0.999(9) * 0.999(9), right?

But 0.999(9) * 0.999(9) cannot be greater than 0.999(9), right?

So, can you tell me what the second digit after the decimal point of of 0.999(9) * 0.999(9) is?

 

[JSuarez]

This is a (very) small ambiguity in positional systems, but first, to answer your question:

How can.9999999999999999999999999999999999999999999999 999999999(repeating forever) multiplied by 2, equal 2?

$$0.9\left(9\right) = \sum^{+\infty}_{n=1}\frac{9}{10^n}=9\frac{0.1}{1-0.1}=1$$

Therefore:

$$2\times0.9\left(9\right) = 2\sum^{+\infty}_{n=1}\frac{9}{10^n}=18\frac{0.1}{1-0.1}=2$$

But this is a general feature of any positional numerical system: for example, $0.1\left(1\right) = 1$ in base 2. In general, given a base b > 1, its highest digit is d = b - 1 and you have:

$$0.d\left(d\right) = 1$$

Note that 1 has the same representation relative to every basis > 1.

 

[Monimonika]

I know, and conform to the ideal that .9 infinitely repeating does in fact equal 1 but I have a thought that keeps bothering me.
How can.9999999999999999999999999999999999999999999999999999999(repeating forever) multiplied by 2, equal 2?

Allow me to demonstrate on how the multiplication can be done.

Now, traditionally when multiplying the following (the *s are used for space-filling, please adjust fonts or copy-paste into a simple text editor to line up characters) (I'm a newbie and can't figure out the LaTeX codes that would make this look even semi-coherent):

*0.999
*x 2
*------

we were taught to start from the right and work our way left:

*0.999
*x 2
*------
*0.018
*0.18
+1.8
--------
*1.998

However, since "2 x 0.999" is really just "2 x (0.009 + 0.09 + 0.9)", we can use the commutative property of addition/multiplication to change the order to "2 x (0.9 + 0.09 + 0.009)" and still get the same answer.

Therefore we can calculate from left to right, instead:

*0.999
*x 2
*------
*1.8
*0.18
+0.018
--------
*1.998

Using this method, we can multiply 0.999... by 2 and get the following:

*0.999999999999...
*x 2
*-----------------
*1.8
*0.18
*0.018
*0.0018
*0.00018
*0.000018
*0.0000018
*0.00000018
..
...
...
+
------------------
*1.999999999999999...

2 x 0.999... = 1.999... = 1 + 0.999... = 1 + 1 = 2Now, onto 0.999... x 0.999... (in my next comment)!

 

[Monimonika]

0.99 * 0.99 is less than 0.99(9) * 0.99(9), right?

But 0.99(9) * 0.99(9) cannot be greater than 0.99(9), right?

So, can you tell me what the first digit after the decimal point of 0.99(9) * 0.99(9) is?

0.999 * 0.999 is less than 0.999(9) * 0.999(9), right?

But 0.999(9) * 0.999(9) cannot be greater than 0.999(9), right?

So, can you tell me what the second digit after the decimal point of of 0.999(9) * 0.999(9) is?

Here is a (long and convoluted) way to multiply 0.999... by 0.999...:

**0.999...
x 0.999...
-------------

I will break this into separate components that will all be added back together at the end. Since 0.999... can be expressed as the geometric series "0.9 + 0.09 + 0.009 + 0.0009 +...", let's multiply each of those components by 0.999...:

**0.999...
x 0.900...
-------------
*0.81
*0.081
*0.0081
*0.00081
*0.000081
.
..
...
+
------------------
*0.8999999999...

and

**0.9999...
x 0.0900...
-------------
*0.081
*0.0081
*0.00081
*0.000081
*0.0000081
.
..
...
+
------------------
*0.08999999999...

and

**0.99999...
x 0.00900...
-------------
*0.0081
*0.00081
*0.000081
*0.0000081
*0.00000081
.
..
...
+
------------------
*0.008999999999...

and so on and so forth...


Yeah, I hope you recognize how the pattern goes. Anyway, let's try adding the multiplied components up:

*0.89999999...
*0.08999999...
*0.00899999...
*0.00089999...
*0.00008999...
.
..
...
+
--------------------

Eeek! Guess I'll have to break even this one down a bit. Remember, the above is the result of "0.999... x (0.9 + 0.09 + 0.009 +...)".

Let's add one row at a time, shall we? Here are the first two rows added together:

*0.89999999...
+0.08999999...
------------------

Let's see...since we know that there are only 9s from the 1000th place onward to the right for both rows, we can reaarange this from "(0.89 + 0.00999...) + (0.08 + 0.00999...)" to "(0.89 + 0.08) + (0.00999... + 0.00999...)".

So, broken down:

*0.89
+0.08
--------
*0.97

plus

*0.00999...
+0.00999...
------------
*0.01999... = 0.00999... x 2

(Remember the "0.999... x 2" thing? Same stuff.)

And then we add the two results together:

*0.97000000...
+0.01999999...
-------------
*0.98999999...

Phew! First two rows done. Now to add the third row (look back if you've lost track of what we were originally doing):

*0.98999999...
+0.00899999...
---------------

Okay, repeat of what we did in the previous steps, except this time it's from the 10000th place:

*0.989
+0.008
---------
*0.997

and

*0.000999...
+0.000999...
------------
*0.001999... = 0.000999... x 2

Which added together would be:

*0.99700000...
+0.00199999...
-------------
*0.99899999...


Adding fourth row:

*0.99899999...
+0.00089999...
---------------

You should know the procedure by now:

*0.9989
+0.0008
---------
*0.9997

and

*0.00009999...
+0.00009999...
------------
*0.00019999... = 0.0000999... x 2

Which added together would be:

*0.99970000...
+0.00919999...
-------------
*0.99989999...


Fifth row:

*0.99989999...
+0.00008999...
---------------
*0.99998999...

Okay, so I skipped typing out the work here, but this comment is getting way too long. Just to refresh your (and my) memory, this is what the 0.999... x 0.999... equation looked like when converted it into an addition equation:

*0.89999999...
*0.08999999...
*0.00899999...
*0.00089999...
*0.00008999...
.
..
...
+0.00000000...
--------------------

I then started adding up the rows one by one (albeit I started by adding the first two rows). The results of each addition are as follows:

1st row: 0.89999999...
2nd row added: 0.98999999...
3rd row added: 0.99899999...
4th row added: 0.99989999...
5th row added: 0.99998999...
6th row added: 0.99999899...
etc.

Well, the pattern should be obvious here, with that "8" going further to the right with the addition of each row. And since this pattern will go on without end, the result would be:

*0.89999999...
*0.08999999...
*0.00899999...
*0.00089999...
*0.00008999...
.
..
...
+
--------------------
*0.99999999...

And thus I show that 0.999... x 0.999... = 0.999...! Go me!

 

[Borek]

The more I look, the more nines I see. I wonder if that means you are all my creation.

 

[Tinyboss]

I know, and conform to the ideal that .9 infinitely repeating does in fact equal 1 but I have a thought that keeps bothering me.
How can.9999999999999999999999999999999999999999999999999999999(repeating forever) multiplied by 2, equal 2?

Dodo's right: if you believe that .(9)=1, then 2*.(9)=2*1=2, end of story. That's the only answer you need...anything else just boils down to more arguments that .(9)=1, which you claim to believe already.

 

[ramsey2879]

n*2 -n = n
.9(9) * 2 -.9(9) = 1.9(9) -.9(9) = 1 = .9(9)

 

[HallsofIvy]

By the definition of our decimal number system, .99999... means the limit of .9, .99, .999, .9999, ...

Multiply each of those by 2:
1.8, 1.98, 1.998, 1.9998, ...

The limit of that is 2.

 

[hoodle]

CjStaal wrote "I know, and conform to the ideal that .9 infinitely repeating does in fact equal 1 but I have a thought that keeps bothering me. How can.999...(repeating forever) multiplied by 2, equal 2?"

It can't, and doesn't. Simple. Why do you conform to the ideal, since it is clearly false? You've noticed the obvious. Those values aren't equal. It is illogical to pretend they are equivalent. Calculator programmers know the answer is wrong even though it's close to the right one, and they therefore program calculators to return a 1, which is the right answer, instead of the .999... that is actually calculated.

Decimal fractions are good for giving us fast close estimates in most cases, but this system is incapable of accurately representing most fractional values. Specifically, any values that are not evenly divisible by ten or its factors of two and five, are to a greater or lesser extent inaccurately represented in decimal fraction format. Some inaccurately represented values terminate, but are in the wrong "place value" positions; the others are the endless repeating or non-repeating representations for values. These "infinite" representations of values are clear indications of a falsely represented value. The value is exact and finite. The attempted representation for it is not, and can never be accurately arrived at through this calculation technique.

The more use of decimal fractions, the greater the likelihood of increased error and deviation in representing any values which include fractions. Those who use them a lot, like engineers, and some scientists, etc. know this, and adjust their decimal fraction calculation to be closer to accurate. There are established techniques for doing this. This still doesn't necessarily fully correct the value representation, rendering it accurate, however. The calculation to correct the error inherent in the decimal fraction calculation system would be more complex, and more work than just doing the calculations accurately in the first place, which can be done using "vulgar" old fashioned fractions.

The decimal fraction system is a buggy computing system, but useful enough for quickly estimating values that are pretty close to the value sought, if that's all that's required. If accuracy is important, old fashioned "vulgar" fractions will do the job with perfect accuracy, but are far more difficult and time-consuming to use. Also, the accurate statement of some important values remains unknown, like pi, for instance. We only have the inaccurate estimate that includes a decimal fraction. This presents difficulties and an impediment, but hopefully some diligent mathematicians will apply themselves to the problem.

Sorry if I'm tardy for the party, but hope this helps.

 

[Borek]

It can't, and doesn't.

Yes it does. 0.(9) = 1, It can be proven in zillion ways. Search the forums for proofs as they were posted on many occasions. The key is in word "forever".

 

[HallsofIvy]

CjStaal wrote "I know, and conform to the ideal that .9 infinitely repeating does in fact equal 1 but I have a thought that keeps bothering me. How can.999...(repeating forever) multiplied by 2, equal 2?"

It can't, and doesn't. Simple. Why do you conform to the ideal, since it is clearly false? You've noticed the obvious. Those values aren't equal. It is illogical to pretend they are equivalent. Calculator programmers know the answer is wrong even though it's close to the right one, and they therefore program calculators to return a 1, which is the right answer, instead of the .999... that is actually calculated.

Decimal fractions are good for giving us fast close estimates in most cases, but this system is incapable of accurately representing most fractional values. Specifically, any values that are not evenly divisible by ten or its factors of two and five, are to a greater or lesser extent inaccurately represented in decimal fraction format. Some inaccurately represented values terminate, but are in the wrong "place value" positions; the others are the endless repeating or non-repeating representations for values. These "infinite" representations of values are clear indications of a falsely represented value. The value is exact and finite. The attempted representation for it is not, and can never be accurately arrived at through this calculation technique.

The more use of decimal fractions, the greater the likelihood of increased error and deviation in representing any values which include fractions. Those who use them a lot, like engineers, and some scientists, etc. know this, and adjust their decimal fraction calculation to be closer to accurate. There are established techniques for doing this. This still doesn't necessarily fully correct the value representation, rendering it accurate, however. The calculation to correct the error inherent in the decimal fraction calculation system would be more complex, and more work than just doing the calculations accurately in the first place, which can be done using "vulgar" old fashioned fractions.

The decimal fraction system is a buggy computing system, but useful enough for quickly estimating values that are pretty close to the value sought, if that's all that's required. If accuracy is important, old fashioned "vulgar" fractions will do the job with perfect accuracy, but are far more difficult and time-consuming to use. We only have the inaccurate estimate that includes a decimal fraction. This presents difficulties and an impediment, but hopefully some diligent mathematicians will apply themselves to the problem.

Sorry if I'm tardy for the party, but hope this helps.

No, it doesn't help at all. Pretty much everything you have said is wrong. "Diligent mathematicians" have applied themselves to the problem (hundreds of year ago) and, in particular, defined exactly what is meant by an "infinite series" and how to calculate them.

In particular, "0.999999.."= $.9+ .9(.1)+ .9(.01)+...= .9(1+ .1+ .1^2+ .1^3+ ...)$ is a "geometric series" which has sum
$$\frac{.9}{1- .1}= \frac{.9}{.9}= 1$$
exactly.

I suggest you learn some mathematics before you make statements like "Also, the accurate statement of some important values remains unknown, like pi, for instance." We know exactly what pi is equal to, thank you. The fact that it cannot be written in decimal form with a finite number of decimal places is irrelevant. The decimal numeration system is only one way of writing numbers- and not a particulary important one.

 

[Count Iblis]

Multiplication by two is a continuous mapping. To get something less trivial (or ambiguous), you need to consider a mapping that is not continuous. E.g., define the function

f(x) = x for |x|<1,

f(1) = 0

Then what is f(0.999999999999999999999999999...) ?


This of course depends on whether you should take the limit first and then evaluate the function or the other way around. My opinion is that it is not natural to take the limit inside the argument of the function first.

 

[Hurkyl]

My opinion is that it is not natural to take the limit inside the argument of the function first.

Why not? That's what you wrote. If you meant something else, you should have written something else!

I've seen this notation:

f(1^-)​

when someone wants the value of f(x) as x approaches 1 from the left, rather than the value of f at 1. Such ideas can even be built into a number system (although it's somewhat subtle and I don't remember how exactly it works). Of course, you could replace use hyperreals or operator algebras to get the effect too.

 

[hoodle]

Borek wrote: "Yes it does. 0.(9) = 1, It can be proven in zillion ways. Search the forums for proofs as they were posted on many occasions. The key is in word 'forever.'"

Thanks for your response. Perhaps you could recommend how to go about the search you suggest. There are many thousands of posts of varying content and titles.

Concerning the key "forever," in the current case it would seem to be equivalent to "never," wouldn't it? That being the case I don't see how any proofs could be logically acceptable, actually. They would only be acceptable if we pretend that something that never happens and can't ever and doesn't ever exist, is equivalent to something that either occurs, or has the potential to occur and be actual.

There's "forever" a little bit missing from every iteration of the calculated value, and never a point when the value actually recoups the missing bit, rendering the value of one. An infinity of infinities can't correct that. There's never a time or an occurrence of the value, in all infinity, when it possesses that little missing bit. Do fairies or angels add that little missing bit somewhere in eternity, since infinity forever remains lacking it? Where in any of the proofs is that missing bit added, exactly, rendering the value valid as an exact equivalency for one?

It seems to me that the point of numbers is they quantify. That's why the number system starts with an absolute quantum, that being represented by the number one, and the value of each and every other number is logically and consistently derived from, and therefore divisible by that value.

If quantity has no consistent meaning, and could be some particular thing, except when we feel it might be fun for it to be something else, then what's the point of quantification through use of numbers? If 0.(9) can equal 1, that system has no valid quantum.

Having no valid and absolute quantum for a constructive basis invalidates the entire number system, as a system of quantification, which is its purpose. None of the values in such a system can be considered valid. Or do you opine otherwise? Is violation of the quantum, and a variable quantum (and therefore random and variable numeration) perfectly okay, for a number system of consistently quantized values?

 

[Borek]

Built in search is not as effective as Google, but this search yields:

https://www.physicsforums.com/showthread.php?t=66671

https://www.physicsforums.com/showthread.php?t=124382

https://www.physicsforums.com/showthread.php?t=47827

https://www.physicsforums.com/showthread.php?t=2849

https://www.physicsforums.com/showthread.php?t=215017

"Forever" works, stating that it doesn't you are repeating ancient Greek arguments (Zeno paradox), long shown to be incorrect. And I have a feeling you are mistaking number representation and its value - these are not the same. Pi is precisely defined, even if we can't write its decimal representation. Same about 1/3. 3*1/3 = 1, even if the decimal representation (nothing else but 0.(9)) seems to say different.

 

[zgozvrm]

I think the easiest way to prove that 0.99999... = 1 is:

Let X = 0.999999...
Then 10X = 9.99999... (moving the decimal point one place to the right)
So 10X - X = 9.999999... - X (subtracting the same number from both sides of the equation)
We then get 9X = 9.99999... - 0.99999...
but 9.99999... = 9 + 0.99999...
so 9X = 9
and X = 1

Also, you can show that
1/9 = 0.11111...
2/9 = 0.22222...
3/9 = 0.33333...
4/9 = 0.44444...
5/9 = 0.55555...
6/9 = 0.66666...
7/9 = 0.77777...
8/9 = 0.88888...
9/9 = 0.99999...

Further,
1/3 * 3 = 1
but 1/3 = 0.33333...
so 0.33333... * 3 = 0.99999... = 1

 

[hoodle]

Dear zgozvrm:

Thanks for your response. My post was removed, so I stopped visiting the site since my honest logical discussion of this fairly simple subject matter was censored, and I am surprised to receive your response. It's a sad day when dogma is enforced as a substitute for logic and discussion of it in fields of math or science, especially at a venue supposedly devoted to teaching and learning in such fields. A good teacher should consider, learn, and be responsive to students and their communications, not merely require recitations of what they insist others thoughtlessly recite. Otherwise, how are teachers and experts any better than mere access to a textbook? No wonder the United States is falling below nearly every other developed country in these fields.

I appreciate your response but find your proof lacking. It's no proof because one of the values is never actually stated. I'll demonstrate.

Let's restate and more clearly identify the value to scrutinize the proof more closely. According to your proof I could use any decimal fraction value really, that is less than one, to represent the value one, and it should be justified since we merely never address the value of the missing portion of the decimal fraction that remains unstated. Let's review. I'll restate your proof differently for comparison:

First, whether we write 0.99999... or 0.9... it can be considered to represent the same assumed untruncated digitally extending value supposedly. To simplify let's use the latter. Now, let's restate your proof, which I find is not a proof but remains an unsolved problem. We still need to solve for the value Y below, which you have neither stated nor solved for.

Let X = 0.9 + ( an as-yet unidentified value "..." that we'll call Y)
Then 10X = 9 + 10Y
So 10X - X = ( 9 - 0.9) + (10Y - Y)
so 9X = 8.1 + 9Y
and (again) X = .9 + Y
Therefore, Y has to equal 1/10 exactly, otherwise stated as 0.1 for X to be equal to 1.

The value of 1/10 stated decimally truncates, is closed-ended and exact. But it isn't exact and doesn't truncate, in your proof. Instead, your supposed proof fails to state the value at all. It merely states "..." instead of an honest value "Y" which needs to be solved for. In fact, at whatever point we truncate the decimal fraction to solve for Y, we will have a differing value for Y, since the trailing digital infinity is an artifact of an imperfect calculating system, and has nothing to do with the values being calculated themselves. We still don't know the value of Y, because it still has not been stated, and no operation we have performed serves to do anything to affix and reveal it as any actual and distinct value. We could use the value .1 + Y or 0.111...= X and still know as little about the value we're assuming for X as when we use 0.9... The proof fails. It is a sophistry, since we must pretend a value has been stated and solved for, when it has not.

Also, 9/9 is not equal to 0.999... It is only approximately equal to it, just as 99 is only approximately equal to 100. It's inexact. The other decimal values you listed are similarly approximations, not exact values. 9/9 is exactly equal to one, and only approximately represented by 0.999... I could say 0.1... = 1 under your proof pretext and it should be just as valid as your statement that 0.9... = 1 since the value of ... has not and never will be actually stated or revealed as any particular value by anyone, ever. A minority of fractional values are accurately and exactly represented by decimal fractions. Most are, instead, approximate representations for the value. This is useful for some purposes, but inexact and sometimes worse than no answer at all when science or math require exact values and measures.

Now, if you believe that 0.11111111... or 0...1... or 0.0999... are equal to .1, or that 1/9 = 1/10, or that 99=100 there are those who might like to show you a bridge for sale. ;-) Thanks, though, for considering the subject matter and offering a response.

-------

For reference, you wrote:

I think the easiest way to prove that 0.99999... = 1 is:

Let X = 0.999999...
Then 10X = 9.99999... (moving the decimal point one place to the right)
So 10X - X = 9.999999... - X (subtracting the same number from both sides of the equation)
We then get 9X = 9.99999... - 0.99999...=
but 9.99999... = 9 + 0.99999...
so 9X = 9
and X = 1

Also, you can show that
1/9 = 0.11111...
2/9 = 0.22222...
3/9 = 0.33333...
4/9 = 0.44444...
5/9 = 0.55555...
6/9 = 0.66666...
7/9 = 0.77777...
8/9 = 0.88888...
9/9 = 0.99999...

Further,
1/3 * 3 = 1
but 1/3 = 0.33333...
so 0.33333... * 3 = 0.99999... = 1
***************

 

[Hurkyl]

Even the greatest teacher cannot teach someone who is unwilling to learn.

You have one chance to convince me that you truly and genuinely do not understand that the notations 0.(9) and 0.999... are being used to notate a decimal with a '9' in every decimal position to the right of the decimal point, and were instead misinterpreting it as being an unspecified, but finite number of '9's.

Otherwise, I'm going to give you a misinformation infraction for preaching dogma in a subject you do not really appear to know much about.



Incidentally, non-standard analysis is no different than standard analysis -- in both of them, 0.999... = 1.

The difference is that, from the viewpoint of standard analysis, non-standard decimals have more positions -- so there exist non-standard decimals that have finitely many 9's by the non-standard measure, but have infinitely many nines by the standard measure, and so are infinitessimally different from 1, but unequal to 1.

 

[hoodle]

Dear Hurkyl:

You said: Even the greatest teacher cannot teach someone who is unwilling to learn. I say, Amen.

You also said: You have one chance to convince me that you truly and genuinely do not understand that the notations 0.(9) and 0.999... are being used to notate a decimal with a '9' in every decimal position to the right of the decimal point, and were instead misinterpreting it as being an unspecified, but finite number of '9's. Otherwise, I'm going to give you a misinformation infraction for preaching dogma in a subject you do not really appear to know much about.

First, if we already know much, why bother talking to you? Isn't this forum for interaction, interchange and learning exchanges between people of differing levels of expertise on the subjects? I am not preaching, I am questioning and investigating, making assertions in that pursuit, and interested in further discussion with honest souls who are similarly interested in the subject matter and its veracity. If you are truly a mathematician or scientist I would expect you to welcome that, even encourage it, and do it yourself too, not dismiss and punish or shun and isolate those who do it. That's what we should all be doing in these fields, whether novice or expert, isn't it? If students are not allowed to make errors, and to arrive at or investigate wrong conclusions, neither do they have the necessary freedom to arrive at the right ones. In that case they are merely memorizers and reciters regurgitating others' ideas thoughtlessly. We would do better just listening to a recording device stating back our own ideas, than wasting others' time getting them to do what any digital recorder does better than a person. With no possibility to investigate and present "misinformation" for scrutiny and testing, then neither do we have the opportunity of developing capability to sort out true things from false, determining for ourselves what then is information and misinformation and thus, doing what we call "learning."

I dispute that the notation indicated represents any particular actual value, however many digits representing nine we write, whether supposedly infinite or not. I assert it pretends to be a representation for a value that has not and cannot be either computed or represented through decimal fraction representation at all, because the decimal fraction system itself is incapable of either calculating or representing that value through any means, even though its attempt to do so is infinite and we are wrongly instructed to pretend it is a value instead of scrutinizing whether it qualifies as one, actually, and what value that would be, if it were a value.

Your question however, deflects from the content of what I provided. Is there a flaw in what I presented? If so, state it clearly, and show work that reveals why. If you really are an expert, I should expect, and will offer you not only one chance, but as many as you need, to simply and directly discuss the simple ideas I provided in as much detail as is necessary to make your case. In particular, do you dispute that construing the problem as I did was invalid? For instance, was it inaccurate to state 0.9... may be converted into 0.9 + (the value of the remainder of the decimal fraction representation for the supposed value)? If so, why? And if not, what's your difficulty with what I offered? If you prefer to issue an "infraction" instead of addressing the subject matter itself, I will assume you are incapabable of discussing the subject freshly and I will question the veracity of your assertions and whether you are actually an expert, since thus far you haven't addressed the subject matter in any substantive way.

I will be just as happy to discover I am wrong or right in this matter, but consider it unfair to close down the discussion, especially on subject matter so simple even young children without higher levels of instruction can easily consider the ideas. Arguing it's too complicated for others to understand, does nothing to support your position. Your response to my investigation is similar to techniques used widely in the dark ages, when authorities attacked those who asked questions and made investigations, rather than addressing or allowing address of the value of the assertions made, when differing ideas were presented than the authoritative position. Do you fancy that experts, even if they agree among themselves, are always right merely by virtue of the fact they agree among themselves, or should they actually have to provide a basis or argument in favor of the ideas they might insist others must adopt, when challenged? The proof fails, and I've indicated why I think so. Please address what I offered rather than attacking and punishing me for offering discussion of the supposed proof. I will be happy to try to learn, if you are able and willing to teach what I have failed to grasp in this matter.

 

[ramsey2879]

Dear zgozvrm:

Thanks for your response. My post was removed, so I stopped visiting the site since my honest logical discussion of this fairly simple subject matter was censored, and I am surprised to receive your response. It's a sad day when dogma is enforced as a substitute for logic and discussion of it in fields of math or science, especially at a venue supposedly devoted to teaching and learning in such fields. A good teacher should consider, learn, and be responsive to students and their communications, not merely require recitations of what they insist others thoughtlessly recite. Otherwise, how are teachers and experts any better than mere access to a textbook? No wonder the United States is falling below nearly every other developed country in these fields.

I appreciate your response but find your proof lacking. It's no proof because one of the values is never actually stated. I'll demonstrate.

Let's restate and more clearly identify the value to scrutinize the proof more closely. According to your proof I could use any decimal fraction value really, that is less than one, to represent the value one, and it should be justified since we merely never address the value of the missing portion of the decimal fraction that remains unstated. Let's review. I'll restate your proof differently for comparison:

First, whether we write 0.99999... or 0.9... it can be considered to represent the same assumed untruncated digitally extending value supposedly. To simplify let's use the latter. Now, let's restate your proof, which I find is not a proof but remains an unsolved problem. We still need to solve for the value Y below, which you have neither stated nor solved for.

Let X = 0.9 + ( an as-yet unidentified value "..." that we'll call Y)
Then 10X = 9 + 10Y
So 10X - X = ( 9 - 0.9) + (10Y - Y)
so 9X = 8.1 + 9Y
and (again) X = .9 + Y
Therefore, Y has to equal 1/10 exactly, otherwise stated as 0.1 for X to be equal to 1.

**

Dear Hoodle Thanks for your input again.
I appreciate that you admit that Y must equal .1 for X to equal 1. I given this careful thought. If Y = .099999... Then 10 Y = .999999... since to multiply a decimal number by 10 you just move the decimal point one place to the right. Note that I did not write 10Y as being equal to .99990... because that doesn't reflect the fact that the 9's are never ending. So now we have 10Y = .9 + Y from which we can deduce that 9Y = .9 or Y = .1
Your logic was correct, you just did not follow it through to the end result. Note we now have X = .99999... = .9 + .1 = 1. If you do not accept this, tell me where this is wrong and cite your authority.
Also, please explain why you hold that .99999... is not equal to 3/3 as was shown by the poster just prior to your post.

 

[Mark44]

Also, 9/9 is not equal to 0.999... It is only approximately equal to it, just as 99 is only approximately equal to 100. It's inexact.

Since you are asserting that 9/9 (i.e., 1) is different from 0.999..., the difference 1 - 0.999... should be nonzero.

By how much do 1 and 0.999... differ?

Sure, 99 and 100 are approximately equal, but no one in his right mind would state that they are equal, since they clearly differ by 1.

The other decimal values you listed are similarly approximations, not exact values. 9/9 is exactly equal to one, and only approximately represented by 0.999... I could say 0.1... = 1

You are missing something fundamental in this discussion.
1 is only approximately represented by .9, .99, and .999, but we are talking about a number with an infinite number of 9s following the decimal point, not a finite number of them, as there are in .9, .99, and .999.

Also, no one is asserting that 0.1111... is equal to 1.
Now, if you believe that 0.11111111... or 0...1... or 0.0999... are equal to .1, or that 1/9 = 1/10, or that 99=100 there are those who might like to show you a bridge for sale. ;-) Thanks, though, for considering the subject matter and offering a response.
[/quote]

 

[Mark44]

hoodle,
I picked out a few paragraphs from some of your posts in this thread that deserve comment over an above what has already been said..

It can't, and doesn't. Simple. Why do you conform to the ideal, since it is clearly false? You've noticed the obvious. Those values aren't equal. It is illogical to pretend they are equivalent. Calculator programmers know the answer is wrong even though it's close to the right one, and they therefore program calculators to return a 1, which is the right answer, instead of the .999... that is actually calculated.

Calculator programmers have memory constraints to contend with. They can't represent 1/9, say, as a decimal fraction with an infinite number of 1s. If a calculation results in a value such as .9999 ...9, with a specified (and finite) number of 9s, some calculators use algorithms to guess what the correct value should be. I seem to remember some of the TI calculators doing this back about 20 years ago, but haven't really followed this technology much since.

Decimal fractions are good for giving us fast close estimates in most cases, but this system is incapable of accurately representing most fractional values. Specifically, any values that are not evenly divisible by ten or its factors of two and five, are to a greater or lesser extent inaccurately represented in decimal fraction format. Some inaccurately represented values terminate, but are in the wrong "place value" positions; the others are the endless repeating or non-repeating representations for values. These "infinite" representations of values are clear indications of a falsely represented value. The value is exact and finite. The attempted representation for it is not, and can never be accurately arrived at through this calculation technique.

What do you mean "Some inaccurately represented values terminate, but are in the wrong "place value" positions"? I have no idea what you're saying here.

As for infinite representations, are you saying that .16666(6) falsely repreresents 1/6? Note that the parenthesized part indicates infinitely repeating 6s.

Concerning the key "forever," in the current case it would seem to be equivalent to "never," wouldn't it? That being the case I don't see how any proofs could be logically acceptable, actually. They would only be acceptable if we pretend that something that never happens and can't ever and doesn't ever exist, is equivalent to something that either occurs, or has the potential to occur and be actual.

How do you interpret forever and never as being equivalent? This would seem to be somewhat out of context, but your response was to a post by Borek (#17), who made an unequivocal statement about the digits in the decimal fraction representation repeating forever. That seems pretty clear to me.

It seems to me that the point of numbers is they quantify. That's why the number system starts with an absolute quantum, that being represented by the number one, and the value of each and every other number is logically and consistently derived from, and therefore divisible by that value.

How is this relevant to the discussion in this thread?

 

[Hurkyl]

I am not preaching, I am questioning and investigating, making assertions in that pursuit, and interested in further discussion with honest souls who are similarly interested in the subject matter and its veracity.

I might have believed you if you, y'know, asked questions, or even gave the slightest hint that you were being speculative with your assertions, rather than trying to speak with authority.

Asking "how can we be sure that 0.(9) refers to a real^* number?" would make you sound like someone willing to learn. "0.(9) doesn't represent any value at all and we are wrongly instructed to pretend otherwise" makes you sound like a crackpot who thinks he's in possession of the Truth and is tilting at a perceived giant global conspiracy.

That said, you sound like a crackpot who thinks he's in possession of the Truth and is tilting at a perceived giant global conspiracy, so thread closed.

*: This is a technical mathematical term. Don't confuse it with its English homonym.

(P.S. the proof of zgozvrm's that you've been talking about is meant for those who have already learned how to do arithmetic in decimal notation and are familiar with the basic algebraic property of the real numbers)